Metric learning

train <- readRDS(here::here("data/spdemand_1id336tow_train.rds"))
# Parameters fixed
x <- train
N <- nrow(train)
s <- 2
k <- 20
method <- "annIsomap"
annmethod <- "kdtree"
distance <- "euclidean"
treetype <- "kd"
searchtype <- "radius" # change searchtype for radius search based on `radius`, or KNN search based on `k`
radius <- .4 # the bandwidth parameter, \sqrt(\elsilon), as in algorithm

metric_isomap <- metricML(x, s = s, k = k, radius = radius, method = method, annmethod = annmethod, eps = 0, distance = distance, treetype = treetype, searchtype = searchtype, invert.h = TRUE)
summary(metric_isomap)

##                Length Class     Mode   
## embedding         672 -none-    numeric
## rmetric          1344 -none-    numeric
## weighted_graph     10 igraph    list   
## adj_matrix     112896 dgCMatrix S4     
## laplacian      112896 dgCMatrix S4

Variable kernel density estimate

For multivariate data, the variable kernel density estimate is given by \[\hat{f}(x) = n^{-1} \sum_i K_{H_i} (x - X_i).\]

# fixed bandwidth
fn <- metric_isomap$embedding
E1 <- fn[,1] # rename as Ed to match the aesthetics in plot_ellipse()
E2 <- fn[,2]
prob <- c(1, 50, 99)
p_outlier <- hdrscatterplot(E1, E2, levels = prob, noutliers = 20, label = NULL) + 
  plot_ellipse(metric_isomap, add = T, n.plot = 50, scale = 20, 
             color = blues9[5], fill = blues9[5], alpha = 0.2)
p_outlier

Rn <- metric_isomap$rmetric # array
n.grid <- 10
f <- vkde2d(x = fn[,1], y = fn[,2], h = Rn, n = n.grid)
plot_contour(metric_isomap, n.grid = n.grid, scale = 0.1)

source(here::here("R/sources/hdrplotting.R"))
p_outlier_vkde <- plot_outlier(x = metric_isomap, n.grid = 50, prob = prob)

library(patchwork)
(p_outlier + p_outlier_vkde ) + coord_fixed()

t-SNE

x <- train
method <- "tSNE"
perplexity <- 30
theta <- 0.5
metric_tsne <- metricML(x, s = s, k = k, radius = radius, method = method, annmethod = annmethod, eps = 0, distance = distance, treetype = treetype, searchtype = searchtype, perplexity = perplexity, theta = theta, invert.h = TRUE)
summary(metric_tsne)

##                Length Class     Mode   
## embedding         672 -none-    numeric
## rmetric          1344 -none-    numeric
## weighted_graph     10 igraph    list   
## adj_matrix     112896 dgCMatrix S4     
## laplacian      112896 dgCMatrix S4

ftsne <- metric_tsne$embedding
E1 <- ftsne[,1]; E2 <- ftsne[,2]
# plot_embedding(metric_tsne)
plot_ellipse(metric_tsne, n.plot = 50)

plot_contour(metric_tsne, n.grid = 20, scale = 1/8)

p_tsne <- plot_outlier(x = metric_tsne, n.grid = 20, prob = prob, noutliers = 20, scale = 1/8)
p_tsne

hdrscatterplot(E1, E2, kde.package = "ks", noutliers = 20) + 
  plot_ellipse(metric_tsne, n.plot = 50, add = T)

p_outlier_vkde + p_tsne

Compare outliers from Isomap and t-SNE

To compare two manifold learning methods, we calculate the overlap among outliers. To do so, we first rank the outliers from highest to lowest densities. Then calculate the correlation of ranks for both methods.

To show our methods performs better, we expect a higher correlation of ranks. This shows robustness in finding outliers using different manifold learning methods.

Compare high density regions

Similarly, we compare the highest density regions from both methods and expect a similar output in detecing most typical observations.